(wave front sets of propagators of Klein-Gordon equation on Minkowski spacetime)

The wave front set of the various propagators for the Klein-Gordon equation on Minkowski spacetime, regarded, via translation invariance, as distributions in a single variable, are as follows:

  • the causal propagator Δ S\Delta_S (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}) has wave front set all pairs (x,k)(x,k) with xx and kk both on the lightcone:

    WF(Δ S)={(x,k)||x| η 2=0and|k| η 2=0andk0} WF(\Delta_S) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \, k \neq 0 \right\}
  • the Hadamard propagator Δ H\Delta_H (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) has wave front set all pairs (x,k)(x,k) with xx on the light cone and k 0>0k^0 \gt 0:

    WF(Δ H)={(x,k)||x| η 2=0and|k| η 2=0andk 0>0} WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; k^0 \gt 0 \right\}
Proof idea

Regarding the causal propagator:

By prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} the singular support of Δ S\Delta_S is the light cone.

Let bC cp ( p,1)b \in C^\infty_{cp}(\mathbb{R}^{p,1}) be a bump function whose compact support includes the origin.

For a p,1a \in \mathbb{R}^{p,1} a point on the light cone, we need to determine the decay property of the Fourier transform of xb(xa)Δ S(x)x \mapsto b(x-a)\Delta_S(x). This is the convolution of distributions of b^(k)e ik μa μ\hat b(k)e^{i k_\mu a^\mu} with Δ^ S(k)\widehat \Delta_S(k). By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} we have

Δ^ S(k)δ(k μk μ(mc) 2)sgn(k 0). \widehat \Delta_{S}(k) \;\propto\; \delta\left( -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \right) sgn(k_0) \,.

This means that the convolution product is the smearing of the mass shell by b^(k)e ik ua μ\widehat b(k)e^{i k_\u a^\mu}.

Since the mass shell asymptotes to the light cone, and since e ik μa μ=1e^{i k_\mu a^\mu} = 1 for kk on the light cone (given that aa is on the light cone), this implies immediately that all kk on the light cone are in the wave front set at the point aa.

It remains to see that no other wave vctors kk are in the wave front set. But if kk is not on the light cone, the for large constants cc the product ckc k has arbitrary large distance form the light cone. Since b^\widehat b is a rapidly decreasing function, it fllows (..?..) that the convolution of the mass shell with b^\widehat b is rapidly decreasing with distance from the light cone, hence rapidly decreasing along all kk not on the light cone.

Now for the Hadamard propagator:

By def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} its Fourier transform is of the form

Δ^ H(k)δ(k μk μ+m 2)Θ(k 0) \widehat \Delta_H(k) \;\propto\; \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 )

First of all it follows that the singular support is still the light cone, because this means that Δ H\Delta_H is a convolution of distributions of Δ S\Delta_S with Θ^δ\widehat {\Theta} \propto \delta', and this convolution does not increase the singular support (…).

Therefore now same argument as before says that the wave front set consists of wave vectors kk on the light cone, but now due to the step function factor Θ(k 0)\Theta(-k_0) now they must be in only one of the two branches.

Revised on November 22, 2017 14:31:33 by Urs Schreiber (